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Geometry

Geometry. 8.3 Similar Polygons. Goals. Identify similar polygons Find the ratio of similarity between similar figures. Solve problems involving similar figures. B. S. 6. 9. R. A. 9. T. 10. C. 15. 8. V. 12. D. Similar Polygons. ABCD and RSTV are similar polygons:

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Geometry

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  1. Geometry 8.3 Similar Polygons

  2. Goals • Identify similar polygons • Find the ratio of similarity between similar figures. • Solve problems involving similar figures. Geometry 8.3 Similar Polygons

  3. B S 6 9 R A 9 T 10 C 15 8 V 12 D Similar Polygons • ABCD and RSTV are similar polygons: • Corresponding angles are congruent. • Corresponding sides are proportional. 6 Geometry 8.3 Similar Polygons

  4. B S 6 9 R A 9 T 10 C 15 8 V 12 D Similar Polygons • Corresponding angles are congruent: • A  R, B  S, C  T, D  V 6 Geometry 8.3 Similar Polygons

  5. B 9 A 9 C 15 12 D Similar Polygons • Corresponding sides are proportional: S 6 R 6 T 10 8 V Geometry 8.3 Similar Polygons

  6. B S 6 9 R A 9 T 10 C 15 8 V 12 D Similar Polygons • Corr. s  • Sides prop. • ABCD ~ RSTV 6 Geometry 8.3 Similar Polygons

  7. Similar Polygons • Corresponding Angles are congruent. • Corresponding Sides are proportional. • Use the symbol “~” for similar. • To show that two polygons are similar, you must prove both things: angles congruent, sides proportional. Geometry 8.3 Similar Polygons

  8. K S 70 70 J Q R L Similarity Statements • List the congruent angles. • Write the ratios of the corresponding sides. J  Q, K  S, L  R Geometry 8.3 Similar Polygons

  9. Example 3 • Are these figures similar? • Yes • Why? • Corr. angles congruent • Corr. sides proportional. E F 2 1.5 H G 4 N 6 M 4 3 O 8 P Geometry 8.3 Similar Polygons

  10. Write the similarity statements. 3 E F 2 1.5 H G 4 N 6 M 4 3 O EFGH ~ NMPO 8 P Or: HEFG ~ ONMP, GFEH ~ PMNO, EHGF ~ NOPM, etc. Geometry 8.3 Similar Polygons

  11. Scale Factor If two polygons are similar, the ratio of two corresponding sides is called the scale factor. Geometry 8.3 Similar Polygons

  12. K S 70 70 J Q R L Scale Factor • The scale factor of JKL to QSR is 10/5 or 2/1. • The scale factor of QSR to JKL is 5/10 or 1/2. 10 5 Geometry 8.3 Similar Polygons

  13. A 10 B F 5 G 14 14 x y I z H D 4 C Perimeter and Similar Figures • ABCD ~ FGHI • Find the scale factor of ABCD to FGHI. • Find the values of x, y, and z. • Find the perimeter of ABCD and FGHI. • Find the ratio of the perimeters. Geometry 8.3 Similar Polygons

  14. A 10 B F 5 G 14 14 x y I z H D 4 C Perimeter and Similar Figures 2. Find the scale factor from ABCD to FGHI. The only known corresponding sides are AB and FG. Geometry 8.3 Similar Polygons

  15. A 10 B F 5 G 14 14 x y I z H D 4 C Perimeter and Similar Figures 3. Find the values of x, y, and z. Geometry 8.3 Similar Polygons

  16. A 10 B F 5 G 14 14 7 7 I 2 H D 4 C Perimeter and Similar Figures 4. Find the perimeter of ABCD and FGHI. P = 42 P = 21 Geometry 8.3 Similar Polygons

  17. A 10 B F 5 G 14 14 7 7 I 2 H D 4 C Perimeter and Similar Figures 5. Find the ratio of the perimeters. Ratio of perimeters 2:1 Ratio of Similarity 2:1 P = 42 P = 21 Geometry 8.3 Similar Polygons

  18. Theorem 8.1 • If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths and is equal to the similarity ratio. Geometry 8.3 Similar Polygons

  19. Theorem 8.1 Example These figures are similar. Find the perimeter of the smaller one. 20 8 P = 100 P = ? Geometry 8.3 Similar Polygons

  20. Problems to Solve Geometry 8.3 Similar Polygons

  21. Problem 1 Solve for x and y if the triangles are similar. 20 x + 6 8 4 y – 2 6 Geometry 8.3 Similar Polygons

  22. Problem 1 Solution Scale Factor is 20/8 20 x + 6 8 4 y – 2 6 Solve for x Solve for y Geometry 8.3 Similar Polygons

  23. Problem 2 Find x and y if the figures are similar. x + 10 85 100 32 60 24 y 95 Geometry 8.3 Similar Polygons

  24. x + 10 85 100 32 60 24 y 95 Problem 2 Solution Similarity Ratio y = 360 - 100 - 85 - 95 y = 80 Geometry 8.3 Similar Polygons

  25. Problem 3 • ABC ~ RST • AB = 20 • ST = 4 • BC = RS • Find BC and RS. Geometry 8.3 Similar Polygons

  26. A R B C S T Problem 3 Solution • ABC ~ RST • AB = 20 • ST = 4 • BC = RS • Find BC and RS. 20 x 4 x Geometry 8.3 Similar Polygons

  27. A R 20 x B C S 4 T x Problem 3 Solution ABC ~ RST Geometry 8.3 Similar Polygons

  28. Problem 4 You want to print a picture in your camera. You have two sizes of paper for your printer: 4 × 6 and 5 × 7. Does it matter? Will the pictures printed from each size of paper be similar? 4 × 6 Sides not proportional, figures not similar. 5 × 7 Geometry 8.3 Similar Polygons

  29. Problem 5 MNOP has a perimeter of 24. Find the perimeter of QRST if MN = 8 and QR = 12. Geometry 8.3 Similar Polygons

  30. Summary • Two polygons are similar if they have the same shape, but a different size. • If polygons are similar corresponding angles are congruent, and corresponding sides are proportional. • The ratio of any two corresponding sides is the scale factor. • The ratio of the perimeters is equal to the ratio of two corresponding sides. Geometry 8.3 Similar Polygons

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